A130063 -id:A130063 - cp's OEIS frontend

A130062 Nonprime numbers k such that k divides 3^((k+1)/2) - 2^((k+1)/2) - 1.

1, 21, 49, 105, 1729, 2465, 2877, 7305, 10585, 15841, 31021, 31621, 32041, 41041, 46657, 52633, 54145, 75361, 83333, 115921, 126217, 162401, 172081, 211141, 282133, 284649, 294409, 334153, 383161, 399001, 417241, 449065, 488881, 530881

Offset: 1

Alexander Adamchuk, May 05 2007

nonn

The perfect squares in listed terms are a(1) = 1, a(3) = 49 = 7^2, a(13) = 32041 = 179^2 and a(29) = 383161 = 619^2.

Note that primes {7,179,619} are the terms of A130060 or primes in A127074.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A097934 (primes p that divide 3^((p-1)/2) - 2^((p-1)/2)).
Cf. A038876 (primes p such that 6 is a square mod p).
Cf. A127071, A127072, A127073, A127074.
Cf. A130058, A130059, A130061, A130063, A130060, A127074.

Select[ 2*Range[100000]-1, !PrimeQ[ # ] && Mod[ PowerMod[3,(#+1)/2,# ] - PowerMod[2,(#+1)/2,# ] - 1, # ] == 0 & ]

More terms from Ryan Propper, Jan 07 2008

A130060 Primes p such that p^2 divides 3^p - 2^p - 1; or primes in A127074(n).

Original entry on oeis.org

2, 3, 7, 179, 619, 17807

Offset: 1

Alexander Adamchuk, May 05 2007

The prime p divides 3^p - 2^p - 1. Quotients (3^p - 2^p - 1)/p, where p = Prime[n], are listed in A127071. - Alexander Adamchuk, Jan 31 2008

a(7) > 10^9. [From D. S. McNeil, Mar 16 2009]

Cf. A127071, A127072, A127073, A127074 = numbers n such that n^2 divides 3^n - 2^n - 1. Cf. A130058, A130059, A130061, A130062, A130063.

Do[ n=Prime[k]; f=PowerMod[3,n,n^2] - PowerMod[2,n,n^2] - 1; If[ IntegerQ[ f/n^2 ], Print[n] ], {k,1,100000} ]

2 more terms found by Ryan Propper, Jan 01 2008.

Incorrect a(7), a(8) removed by D. S. McNeil, Mar 16 2009. (The old version was 2,3,7,179,619,17807,135433,1376257.)

A130061 Numbers k that divide 3^((k-1)/2) - 2^((k-1)/2) - 1.

Original entry on oeis.org

1, 3, 35, 147, 195, 219, 291, 399, 579, 583, 723, 939, 1011, 1023, 1227, 1299, 1371, 1443, 1731, 1803, 2019, 2307, 2499, 2811, 3003, 3027, 3099, 3387, 3459, 3603, 3747, 3891, 3963, 4467, 4623, 4827, 4971, 5187, 5259, 5331, 5403, 5619, 5979, 6051, 6267

Offset: 1

Alexander Adamchuk, May 05 2007

nonn

It appears that all terms are composite except a(1) = 1 and a(2) = 3. Most listed terms are divisible by 3, except {1, 35, 583, 70643, ...}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A097934 (primes p that divide 3^((p-1)/2) - 2^((p-1)/2)).
Cf. A038876 (primes p such that 6 is a square mod p).
Cf. A127071, A127072, A127073, A127074.
Cf. A130058, A130059, A130060, A130062, A130063.

Select[ Range[10000], Mod[ PowerMod[3,(#-1)/2,# ] - PowerMod[2,(#-1)/2,# ] -1, # ]==0&]

cp's OEIS Frontend

A130062 Nonprime numbers k such that k divides 3^((k+1)/2) - 2^((k+1)/2) - 1.

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A130060 Primes p such that p^2 divides 3^p - 2^p - 1; or primes in A127074(n).

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A130061 Numbers k that divide 3^((k-1)/2) - 2^((k-1)/2) - 1.

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica